Question: For how many bases between two and nine inclusive does the representation of $576_{10}$ have a final digit of 1?
Answer: For 576 to have a final digit of 1 when represented in base $b$, we must have that $576-1$ is divisible by $b$. To see this, note that any integer whose base-$b$ representation ends in 0 is divisible by $b$, just as any integer whose decimal representation ends in 0 is divisible by 10.  Since $575 = 5^2 \cdot 23$, the only base which satisfies the given condition is 5.  Therefore, there is $\boxed{1}$ such base.